(1) Field of the Invention
The present invention relates generally to steady, incompressible fluid flow and more particularly to a method for predicting steady, incompressible fluid flow over a large-scale flow field.
(2) Description of the Prior Art
The prediction of a steady, incompressible fluid flow over a large-scale flow field is an invaluable design tool. A well-known method of achieving such prediction is the solving of the Navier Stokes equations, shown below: ##EQU1## where V=V(u,v,w) is the vector velocity field containing the x, y, and z component of velocity, P is the fluid pressure, .nu. is the kinematic fluid viscosity, and p is the constant fluid density. .gradient. indicates the divergence or gradient of a vector of scaler field respectively, and .gradient..sup.2 indicates the vector Laplacian. The D/Dt expression is the symbol for the substantial derivative and the symbol "t" represents time.
The above-shown two equations represent the expressions for conservation of fluid mass and conservation of fluid momentum for an incompressible fluid, respectively. An incompressible fluid is defined here as a fluid which has constant density everywhere in the domain in question.
Unfortunately, in their proper form, the complexity of these equations makes them analytically and numerically unsolvable for large and/or complicated geometric configurations.
In order to overcome this problem, a method called pseudocompressibility was introduced by A. J. Chorin in "A Numerical Method for Solving Incompressible Viscous Flow Problems", J. Comp. Physics, Vol. 2, pp. 12-16, 1967, to alter the incompressible equations shown above to the form shown in equations (3) and (4) below: ##EQU2## The difference between equations (1) and (2) and (3) and (4) can be seen as the addition of the .differential..rho./.differential.t term in the conservation of fluid mass equation. This term is used to change the mathematical representation of the fluid equations, such that they can be solved easier and more directly. However, by adding the pressure time derivative term, the fluid behavior that these equations describe has been changed. In equations (1) and (2) the "t" symbol represents true physical time. In equation (3) and (4) "t" no longer represents time; it is only a mathematical iteration parameter used to allow the more straight forward solution method. A solution to equations (3) and (4) will provide only steady incompressible flow fields. If the original equations (1) and (2) were solved, unsteady flow fields could also be predicted. Therefore in the following technique described, the resulting flow fields predicted by the method discussed will be solutions for steady incompressible flow fields only. The equations (1) and (2) are known as the incompressible Navier Stokes equations.
Currently, the pseudocompressibility equations are solved for a plurality of points or nodes describing the flow area under consideration. The geometric flow area of concern must first be modeled, as discrete mathematical pieces. These mathematical pieces are the actual sections over which the governing equations are solved. Three very popular methods include the finite element approach, finite volume approach and the finite difference method. In order to better understand the prior art, a brief description is provided of the finite difference method used to solve the pseudocompressibility equations. The finite difference method involves describing the flow area with many small points, called grid points, mesh points, or nodes. FIG. 1(a) shows an example of a two-dimensional fluid flow problem and FIG. 1(b) shows a representative finite difference mesh which might be constructed for this particular problem.
In FIG. 1(a), a two-dimensional flow, indicated by arrow 10, enters a flow area 11 between two walls 13 and 15, respectively, and exits as flow 20. The change in the structure of the flow field as it passes through area 11 is the issue of concern. FIG. 1(b) shows the flow area 11 modeled as a plurality of nodes 17 that define the geometry of flow area 11. It is to be noted that nodes 17 are chosen closer together over areas of changing geometry, i.e., changing flow. It should also be noted that the mesh shown here may not be a good finite difference mesh in actuality, but it is used to illustrate the following discussion.
At all the nodes 17 which fill the flow area 11, the governing pseudocompressibility equations must be solved. The pseudocompressibility equations must be solved at each node 17 for pressure and a three-dimensional velocity vector in order to describe the flow at node 17. For example, if it takes 1,000 points to adequately describe a given geometry, then there are a total of 4,000 equations which must be solved since it takes 4 equations at each node to describe the fluid. The boundary points are chosen such that they accurately describe the physical boundaries of the flow area 11 of interest as well as the important features of the flow field in area 11.
Since the pseudocompressibility equations are not the exact incompressible equations describing the fluid behavior, (i.e., they are only a mathematical contrivance), the solution actually involves an iterative scheme. In other words, solving the equations once for all nodes 17 is insufficient to obtain the final answer. The process begins with an initial estimate of the pressure, and three velocity values at every node 17. The initial estimate of the flow in area 11 is used in the pseudocompressibility equations, and solving the equations then predicts the next best solution to the flow field problem. This initial prediction will not generally satisfy the incompressible Navier Stokes equations. Accordingly, the newly predicted values for the pressures and velocities are then used as the new estimate of the fluid flow in area 11. This new estimate is used in the pseudocompressibility equations and a new flow field is again predicted. This iterative process continues until the solution satisfies the Navier Stokes equations within a desired tolerance. This iteration procedure can take anywhere from several hundred to several thousand cycles through all the equations for all the nodes 17.
Large-scale flow areas further complicate the analysis since the number of nodes required to accurately describe the flow area of concern may number well into the millions. At this point, computer memory limitations require that the flow area be divided into a plurality of blocks. This is a well-known technique in the art and is referred to as multiblocking.
Multiblocking allows the computer to take a large intractable flow field domain, which has been represented by finite difference nodes, and break it up into smaller, manageable regions which can be solved individually. Since computers today have limited amounts of memory that restrict the size problem that can be solved at one time, the multiblocking process provides a way around this. However, breaking up the larger domain into smaller multiblock domains creates artificial internal boundaries in the finite difference mesh. These artificial boundaries occur at multiblock boundaries which occur arbitrarily within the domain and not at the true boundaries of the physical problem. While the breakup into multiblock domains enables the computers to solve the smaller system, the artificial boundaries can affect the accuracy of the final solution as well as significantly increase the number of numerical iterations it takes to obtain the final flow field prediction. This occurs because the Navier Stokes equations cannot be directly solved on the artificial multiblock boundaries. These boundaries really require boundary conditions much the same as the true physical boundaries require specification of boundary conditions. Therefore, the method for treating the artificial multiblock boundaries is the key to obtaining accurate steady, incompressible flow field predictions with a minimal amount of numerical iterations.